<p>In this unit, students investigate real-world applications of parabolas. Students will:<br>- gain conceptual knowledge of each extracted piece of information from a quadratic function.<br>- visualize a given scenario and illustrate the function on paper, defining and explaining each attribute within the problem context.<br>- use quadratic functions to model real-world scenarios.<br>- classify real-world scenarios based on their function and argue the reasoning for the use of such a model.<br>&nbsp;</p>

<p>- How can students represent a given function? What relationships can be drawn between the various representations?<br>- Which function best represents a specific real-world scenario? How does the function manifest itself in the real world?<br>- How can equations, tables, and graphs be utilized to examine the rate of change and other relevant information for a real-world problem or function?</p>

<p>- Quadratic Function: A function in the form of <span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><i>y</i> = a<i>x</i>² + c, or <i>y</i> = a<i>x</i>² + b<i>x</i> + c, where <i>a</i> ≠ 0.</span><br>- Parabola: The shape of a quadratic function.</p>

<p>- graph paper&nbsp;<br>- basketball (if desired)<br>- Lesson 3 Exit Ticket (M-A2-7-3_Lesson 3 Exit Ticket)&nbsp;</p>

<p>- Examine the individual work of selected students to observe their progress in independent and collaborative tasks. Ask students to explain their motivation for completing each task. The quality of responses will reflect the level of involvement with the objectives of each section of the session.&nbsp;<br>- Evaluate student performance on the cumulative PowerPoint exercise in Lesson 3 Exit Ticket (M-A2-7-3_Lesson 3 Exit Ticket), which measures each presenter's ability to teach and understand.<br>&nbsp;</p>

<p>Active Engagement&nbsp;<br>W: The entire session consists of active participation and inquiry; the teacher does not provide any direct guidance. Students are taught to imagine, problem-solve, and connect. Students are given the opportunity to sketch forth concepts and discuss explanations.<br>H: Students are drawn to the lesson since it is open and allows for liberty. The entertaining and thought-provoking activities keep kids interested. How many people have considered falling down a slide as a parabolic function?<br>E: The lecture is divided into two parts: Part 1 explores real-world quadratic functions. Part 2 requires higher-level thinking, with students determining two examples from two different fields, as well as determining a belief and arguing for it.<br>R: The activities and PowerPoint review assignments encourage students to reflect, revisit, revise, and rethink. Students are given responsibility for their own study of quadratics in the actual world.<br>E: Collaborative work, including a cumulative PowerPoint, will give students enough opportunities to assess their understanding.<br>T: With an even split of individual and collaborative exercises, and discussion time at the end of each, all students are afforded help to study, reflect, and engage in real-world quadratic functions.<br>O: The class is open-ended, including both structured and loosely structured activities/explorations. The lesson focuses on and aims to teach abstract thinking.</p>

<p><strong>Part 1</strong></p><p><strong>Activity 1 &nbsp;</strong></p><p>The route of a fighter jet is described by the function h = -15m² + 60,000. Draw the graph of the jet's route and answer the following questions:</p><p>1. After 28 minutes, what is the approximate height? How about 54 minutes?</p><p>2. How long does it take the jet to reach an altitude of 46,000 feet?</p><p>3. What is a vertex, and how does it relate to the problem?</p><p>4. If another fighter plane, represented by the function h = -17m² + 56,000, departs at the same time for the same location, which one will make the quickest trip? Why?</p><p>5. What is the domain and range for each jet path? Explain the concept in daily language, according to the problem's context.</p><p><strong>Activity 2</strong></p><p>A tennis ball is thrown into a trash can. Draw the route of the ball, from the moment it leaves the thrower's hands. Label and describe each step of the process on the graph, which represents the ball's path over time. Provide information about the vertex, domain, and range. Describe and clarify the problem. What are the <i>x</i>- and <i>y</i>- intercepts? What do they mean in the context of the problem?</p><p>Students will compare graphs and explore potential explanations for differences and similarities. Students should discuss the concept of rate of change using quadratic functions. What is the rate of change here? (Encourage students to study the function with a real tennis ball.)</p><p><strong>Part 2</strong></p><p><strong>Activity 3</strong></p><p>Divide the students into groups of three or four. Create two real-world scenarios involving quadratic functions, model them with tables and graphs, and ask and answer at least three appropriate questions about each. Create one problem from the scientific sphere and one from the economic world. Create a graphic to show the class.</p><p><strong>Activity 4</strong></p><p>Monique states that a person moving down a slide represents a parabolic function. Determine if you agree or disagree. Provide supporting evidence, such as illustrations, examples, and explanations.</p><p><strong>Activity 5: Performance</strong></p><p>Divide the students into groups of three or four. Students will build an animated PowerPoint presentation with two purposes. Purpose 1 is to illustrate function transformations. Audio should assist and explain transformations, while animation is used to demonstrate the different transformed functions. Purpose 2 is to show a real-world quadratic function in action. The PowerPoint will introduce a problem, simulate a real-world scenario, represent the function in equation, tabular, and graphical forms, and ask and answer three or four questions.</p><p><strong>Extension:</strong></p><p>Ask students to provide instances of related absolute values and quadratic functions. Students should show each using appropriate representations and explain why they chose the function for each example. In other words, why was an absolute value function chosen to model this section of the example? Could we use the quadratic function? What would happen if we used a quadratic function in our interpretations? (and vice versa.) Encourage students to delve deeply into their understanding of the relationships between absolute value and quadratic functions in the real world. The goal is to see if students can create links between the two types of functions, demonstrating a very high level of mental comprehension about these notions.</p>